7.5 Scale Drawings proportionally? How can you enlarge or reduce a drawing 1 ACTIVITY: Comparing Measurements Work with a partner. The diagram shows a food court at a shopping mall. Each centimeter in the diagram represents 40 meters. a. Find the length and the width of the drawing of the food court. length: cm width: cm b. Find the actual length and width of the food court. Explain how you found your answers. length: m width: m drawing length c. Find the ratios actual length and drawing width. What do you notice? actual width 2 ACTIVITY: Recreating a Drawing Work with a partner. Draw the food court in Activity 1 on the grid paper so that each centimeter represents 20 meters. COMMON CORE Geometry In this lesson, you will use scale drawings to findactualdistances. find scale factors. use scale drawings to find actual perimeters and areas. recreate scale drawings at a different scale. Learning Standard 7.G.1 a. What happens to the size of the drawing? b. Find the length and the width of your drawing. Compare these dimensions to the dimensions of the original drawing in Activity 1. 298 Chapter 7 Constructions and Scale Drawings
3 ACTIVITY: Comparing Measurements Work with a partner. The diagram shows a sketch of a painting. Each unit in the sketch represents 8 inches. a. Find the length and the width of the sketch. length: units width: units b. Find the actual length and width of the painting. Explain how you found your answers. length: in. width: in. sketch length sketch width c. Find the ratios and actual length actual width. What do you notice? Math Practice Specify Units How do you know whether to use feet or units for each measurement? 4 ACTIVITY: Recreating a Drawing Work with a partner. Let each unit in the grid paper represent 2 feet. Now sketch the painting in Activity 3 onto the grid paper. a. What happens to the size of the sketch? b. Find the length and the width of your sketch. Compare these dimensions to the dimensions of the original sketch in Activity 3. 5. IN YOUR OWN WORDS How can you enlarge or reduce a drawing proportionally? 6. Complete the table for both the food court and the painting. Perimeter Area Actual Object Original Drawing Your Drawing Compare the measurements in each table. What conclusions can you make? 7. RESEARCH Look at some maps in your school library or on the Internet. Make a list of the different scales used on the maps. 8. When you view a map on the Internet, how does the scale change when you zoom out? How does the scale change when you zoom in? Use what you learned about enlarging or reducing drawings to complete Exercises 4 7 on page 303. Section 7.5 Scale Drawings 299
7.5 Lesson Lesson Tutorials Key Vocabulary scale drawing, p. 300 scale model, p. 300 scale, p. 300 scale factor, p. 301 Study Tip Scales are written so that the drawing distance comes first in the ratio. Scale Drawings and Models A scale drawing is a proportional, two-dimensional drawing of an object. A scale model is a proportional, three-dimensional model of an object. Scale The measurements in scale drawings and models are proportional to the measurements of the actual object. The scale gives the ratio that compares the measurements of the drawing or model with the actual measurements. 10 mi drawing distance actual distance : 10 mi drawing actual EXAMPLE 1 Finding an Actual Distance What is the actual distance d between Cadillac and Detroit? Step 1: Use a centimeter ruler to find the distance on the map between Cadillac and Detroit. The map distance is about 3.5 centimeters. Step 2: Use the scale to write and solve a proportion. 50 mi = 3.5 cm map distance d mi actual distance d = 50 3.5 d = 175 Cross Products Property Multiply. Escanaba Marquette Traverse City Cadillac Grand Rapids Lansing Kalamazoo Alpena Flint Ann Arbor : 50 mi Saginaw Detroit So, the distance between Cadillac and Detroit is about 175 miles. Exercises 8 11 1. What is the actual distance between Traverse City and Marquette? 300 Chapter 7 Constructions and Scale Drawings
EXAMPLE Crust 2 Finding a Distance in a Model The liquid outer core of Earth is 2300 kilometers thick. A scale model of the layers of Earth has a scale of : 500 km. How thick is the liquid outer core of the model? A 0.2 in. B 4.6 in. C 0.2 km D 4.6 km Liquid outer core Mantle Solid inner core 500 km = x in. 2300 km model thickness actual thickness 500 km 2300 km = x in. 2300 km 2300 km Multiplication Property of Equality 4.6 = x Simplify. So, the liquid outer core of the model is 4.6 inches thick. The correct answer is B. 2. The mantle of Earth is 2900 kilometers thick. How thick is the mantle of the model? A scale can be written without units when the units are the same. A scale without units is called a scale factor. EXAMPLE 3 Finding a Scale Factor A scale model of the Sergeant Floyd Monument is 10 inches tall. The actual monument is 100 feet tall. a. What is the scale of the model? model height 10 in. = actual height 100 ft = 10 ft The scale is : 10 ft. b. What is the scale factor of the model? Write the scale with the same units. Use the fact that 1 ft = 12 in. scale factor = 10 ft = 120 in. = 1 120 The scale factor is 1 : 120. Exercises 12 16 3. A drawing has a scale of 1 mm : 20 cm. What is the scale factor of the drawing? Section 7.5 Scale Drawings 301
EXAMPLE : 2 mm 4 Finding an Actual Perimeter and Area The scale drawing of a computer chip helps you see the individual components on the chip. a. Find the perimeter and the area of the computer chip in the scale drawing. When measured using a centimeter ruler, the scale drawing of the computer chip has a side length of 4 centimeters. So, the perimeter of the computer chip in the scale drawing is 4(4) = 16 centimeters, and the area is 4 2 = 16 square centimeters. b. Find the actual perimeter and area of the computer chip. 2 mm = 4 cm s mm drawing distance actual distance s = 2 4 s = 8 Cross Products Property Multiply. The side length of the actual computer chip is 8 millimeters. So, the actual perimeter of the computer chip is 4(8) = 32 millimeters, and the actual area is 8 2 = 64 square millimeters. Study Tip The ratios tell you that the perimeter of the drawing is 5 times the actual perimeter, and the area of the drawing is 5 2 = 25 times the actual area. drawing perimeter drawing area c. Compare the ratios and actual perimeter actual area to the scale factor. Use the fact that = 10 mm. scale factor = 2 mm 10 mm = 2 mm = 5 1 drawing perimeter 16 cm = actual perimeter 32 mm = 2 mm = 5 1 drawing area 16 cm2 = actual area 64 mm = 2 2 4 mm ( = 2 2 mm) 2 = ( 5 1) 2 So, the ratio of the perimeters is equal to the scale factor, and the ratio of the areas is equal to the square of the scale factor. Exercises 22 and 23 4. WHAT IF? The scale of the drawing of the computer chip is : 3 mm. How do the answers in parts (a) (c) change? Justify your answer. 302 Chapter 7 Constructions and Scale Drawings
7.5 Exercises Help with Homework 1. VOCABULARY Compare and contrast the terms scale and scale factor. 2. CRITICAL THINKING The scale of a drawing is 2 cm : 1 mm. Is the scale drawing larger or smaller than the actual object? Explain. 3. REASONING How would you find the scale factor of a drawing that shows a length of 4 inches when the actual object is 8 feet long? 9+(-6)=3 3+(-3)= 4+(-9)= 9+(-1)= Use the drawing and a centimeter ruler. Each centimeter in the drawing represents 5 feet. 4. What is the actual length of the flower garden? 5. What are the actual dimensions of the rose bed? 6. What are the actual perimeters of the perennial beds? 7. The area of the tulip bed is what percent of the area of the rose bed? Use the map in Example 1 to find the actual distance between the cities. 1 8. Kalamazoo and Ann Arbor 9. Lansing and Flint 10. Grand Rapids and Escanaba 11. Saginaw and Alpena Find the missing dimension. Use the scale factor 1 : 12. perennial bed tulip bed Item Model Actual rose bed perennial bed 2 3 12. 13. 14. 15. 16. Mattress Length: 6.25 in. Length: in. Corvette Length: in. Length: 15 ft Water tower Depth: 32 cm Depth: m Wingspan Width: 5.4 ft Width: yd Football helmet Diameter: mm Diameter: 2 17. ERROR ANALYSIS A scale is : 20 m. Describe and correct the error in finding the actual distance that corresponds to 5 centimeters. 20 m = x m 5 cm x = 0.25 m Section 7.5 Scale Drawings 303
Use a centimeter ruler to measure the segment shown. Find the scale of the drawing. 18. 120 m 19. Iris Cornea Pupil Lens Vitreous humor 24 mm 20. REASONING You know the length and the width of a scale model. What additional information do you need to know to find the scale of the model? 21. OPEN-ENDED You are in charge of creating 16 ft a billboard advertisement with the dimensions shown. a. Choose a product. Then design the billboard using words and a picture. b. What is the scale factor of your design? 8 ft YOUR AD HERE 4 22. CENTRAL PARK Central Park is a rectangular park in New York City. : 320 m a. Find the perimeter and the area of Central Park in the scale drawing. b. Find the actual perimeter and area of Central Park. 23. ICON You are designing an icon for a mobile app. a. Find the perimeter and the area of the icon in the scale drawing. b. Find the actual perimeter and area of the icon. 24. CRITICAL THINKING Use the results of Exercises 22 and 23 to make a conjecture about the relationship between the scale factor of a drawing and the ratios drawing perimeter drawing area and actual perimeter actual area. : 2.5 mm 304 Chapter 7 Constructions and Scale Drawings
Recreate the scale drawing so that it has a scale of : 4 m. 25. 26. : 8 m : 2 m The shuffleboard diagram has a scale of : 1 ft. Find the actual area of the region. 27. red region 28. blue region 29. green region 30. BLUEPRINT In a blueprint, each square has a side length 1 of 4 inch. a. Ceramic tile costs $5 per square foot. How much would it cost to tile the bathroom? b. Carpet costs $18 per square yard. How much would it cost to carpet the bedroom and living room? c. Which has a greater unit cost, the tile or the carpet? Explain. Reduced Drawing of Blueprint Bedroom Bathroom : 16 ft Living room 31. Modeling You are making a scale model of the solar system. The radius of Earth is 6378 kilometers. The radius of the Sun is 695,500 kilometers. Is it reasonable to choose a baseball as a model of Earth? Explain your reasoning. Plot and label the ordered pair in a coordinate plane. (Skills Review Handbook) 32. A( 4, 3) 33. B(2, 6) 34. C(5, 1) 35. D( 3, 7) 36. MULTIPLE CHOICE Which set of numbers is ordered from least to greatest? (Section 6.2) A 7 20, 32%, 0.45 B 17%, 0.21, 3 25 C 0.88, 7 8, 93% D 57%, 11 16, 5.7 Section 7.5 Scale Drawings 305